Products defined in the context of noncommutative gauge theory allow for an interpolation between exact results on tachyon potentials at zero and large background B-fields. Techniques for computations of effective actions are transposed from the framework
solitonswithanomalousdimensionsincludedinthenoncommutativepicture,butontheotherhand,recentstudiesofthetoy-modelofnoncommutativep-adicstring[28,29,30]couldprovideahint.Allowingthecoordinatesontheboundaryoftheworld-sheettotakevaluesinthe eldQpofp-adicnumbersledtoanexactlysolvable eldtheoryforopenstringtachyons.Wearethereforefurnishedwiththee ectiveLagrangianfortachyons.Recentworksonnoncom-mutativesolitons[28]reliedonthefollowingdeformationbystar-productsinducedbyatermcouplingtheworld-sheettoaB- eld.Thisconstructionproceededinawaythatisinverseoftheonewechoseaboveinordinary(Archimedean)stringtheory.Tachyonsofthep-adicstring(orp-tachyons)werenamely rstconsideredas eldsendowedwiththeiranomalousdimen-sions,withnoB- eldturnedon[31].Thee ectiveLagrangianwassubsequentlydeformedbycouplingtheboundaryoftheworld-sheettoamagnetic eld.Wethushavegotatheoryofnoncommutativep-tachyonswithbuilt-inanomalousdimensions.Noncommutativetachyonsofordinarystringwereontheotherhandconsideredinalimitwheretheanomalousdimensionsvanish,andthepresentworkproposedthatanomalousdimensionsshouldbeincorporatedasdeformationsofthenoncommutative eldtheory,withoutdestroyingitsstructure.Thetoy-modelofp-adicstringsisexpectedtoleadtoordinarystringswhenpissentto1.Thislimitwassuccessfullycheckedinthecommutativeset-upbyGerasimovandShatashviliin[16],whereitwasobservedthatthee ectiveactionfromboundarystring eldtheoryandthep→1limitofp-adicstringsleadtothesameequationsofmotionuponachangeofvariables.FurthercheckscouldberelatedtoaninterpretationofBasagenuineclosed-stringbackground,insteadofthemagnetic- eldpicture.
Acknowledgements
IthankMarcosMari no,RubenMinasianandPierreVanhovefordiscussions.ThisworkwaspartiallysupportedbyECExcellenceGrantMEXT-CT-2003-509661.
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